The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open n-manifold (n ≥ 7). This paper may be viewed as a generalization of Siebenmann's famous collaring theorem to open manifolds with non-stable fundamental group systems at infinity. AMS Classification numbers Primary: 57N15, 57Q12Secondary: 57R65, 57Q10

We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete characterization of pseudo-collarability for manifolds of dimension at least 6. 57N15, 57Q12; 57R65, 57Q10

Quillen's famous plus construction plays an important role in many aspects of manifold topology. In our own work on ends of open manifolds, an ability to embed cobordisms provided by the plus construction into the manifolds being studied was a key to completing the main structure theorem [GT2]. In this paper we develop a 'spherical modification' trick which allows for a constructive approach to obtaining those embeddings. More importantly, this approach can be used to obtain more general embedding results. In this paper we develop generalizations of the plus construction (together with the corresponding group-theoretic notions) and show how those cobordisms can be embedded in manifolds satisfying appropriate fundamental group properties. Results obtained here are motivated by, and play an important role in, our ongoing study of noncompact manifolds [GT3].Theorem 1.1 (Embedded Manifold Plus Construction). Let R be a connected manifold of dimension ≥ 6, B be a closed component of ∂R, and K ⊆ ker (π 1 (B) → π 1 (R)) Date: January10, 2012. 1991 Mathematics Subject Classification. Primary 57N15, 57Q12; Secondary 57R65, 57Q10.

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with compact boundary, the condition of inward tameness has substatial implications for the algebraic topology at infinity. In particular, every inward tame manifold with compact boundary has stable homology (in all dimensions) and semistable fundamental group at each of its ends. In contrast, we also construct examples of this sort which fail to have perfectly semistable fundamental group at infinity. In doing so, we exhibit the first known examples of open manifolds that are inward tame and have vanishing Wall finiteness obstruction at infinity, but are not pseudo-collarable. AMS Classification numbers Primary: 57N15, 57Q12Secondary: 57R65, 57Q10

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